The proof uses real interpolation theory. Once again, for background on interpolation theory we refer to the survey of Connes [1]} and the references therein. The main reference for real interpolation theory there is the article of Lions-Peetre [2]} (see also [3]}, [4]}).
Let \(x\in L_{p}(\theta ^n)\) , \(p>2\) . By Theorem REF , for every \(g\in \ell _{p,\infty }(\mathbb {Z}^n)\) , the operator \(\lambda (x)g(-i\nabla )\)  is in \({L}_{p,\infty }\) . Thus, we have a linear operator \(\Phi _x: \ell _{p,\infty }(\mathbb {Z}^n)\rightarrow {L}_{p,\infty }\)  given by
\(\Phi _x(g)=\lambda (x)g(-i\nabla ), \qquad g\in \ell _{p,\infty }.\) 
Furthermore, the Cwikel-type estimate (REF ) gives
\(\Vert  \Phi _x(g)\Vert _{{L}_{p,\infty }} \le c_+(p)\Vert x\Vert _{L_p} \Vert g\Vert _{\ell _{p,\infty }} \qquad \forall g \in \ell _{p,\infty }.\) 
In addition, as \(L_p(n_\theta )\subset L_2(n_\theta )\) , it follows from Theorem REF  that, given any \(q\in (1,2)\) , if \(g\in \ell _{q,\infty }(\mathbb {Z}^n)\) , then \(\Phi _x(g)\in {L}_{q,\infty }\) , and we have
\(\big \Vert  \Phi _x(g)\big \Vert _{{L}_{q,\infty }} \le c_{-}(q) \Vert x\Vert _{L_2} \Vert g\Vert _{\ell _{q,\infty }}\le c_{-}(q) \Vert x\Vert _{L_p} \Vert g\Vert _{\ell _{q,\infty }}.\) 
As \(2\in (q,p)\) , the spaces \(\ell _{2,\infty }(\mathbb {Z}^n)\)  and \({L}_{2,\infty }\)  are real interpolation spaces for the pairs of Banach spaces \((\ell _{q,\infty }(\mathbb {Z}^n),\ell _{p,\infty }(\mathbb {Z}^n))\)  and \(({L}_{q,\infty },{L}_{p,\infty })\)  with the exact same exponents. Namely, in the notation of [1]}, and with the same norm equivalences (see e.g. [6]}), we have
\(\ell _{2,\infty }(\mathbb {Z}^n)=[\ell _{q,\infty }(\mathbb {Z}^n),\ell _{p,\infty }(\mathbb {Z}^n)]_{\theta ,\infty }\)  and \({L}_{2,\infty }=[{L}_{q,\infty },{L}_{p,\infty }]_{\theta ,\infty }\) , with \(\theta \in (0,1)\)  such that \(1/2=(1-\theta )q^{-1}+\theta p^{-1}\)  (see [1]} and [8]}). That is, there are constants \(\gamma _1(p,q)>0\)  and \(\gamma _2(p,q)>0\)  such that
\(\gamma _2(p,q)^{-1}\Vert g\Vert _{\ell _{2,\infty }} \le \Vert g\Vert _{[\ell _{q,\infty },\ell _{p,\infty }]_{\theta ,\infty }} \le \gamma _1(p,q) \Vert g\Vert _{\ell _{2,\infty }}\qquad \forall g\in \ell _{2,\infty }(\mathbb {Z}^n),\\\gamma _2(p,q)^{-1}\Vert T\Vert _{{L}_{2,\infty }} \le \Vert T\Vert _{[{L}_{q,\infty },{L}_{p,\infty }]_{\theta ,\infty }} \le \gamma _1(p,q) \Vert T\Vert _{{L}_{2,\infty }} \qquad \forall T\in {L}_{2,\infty }.\) 
We assume that \(\gamma _1(p,q)\)  and \(\gamma _2(p,q)\)  are the best constants in the above inequalities.
By combining the estimates (REF )–(REF ) with real interpolation theory (see [1]}) and by using the
inequalities (REF )–() shows that \(\Phi _x\)  induces a continuous linear map from \(\ell _{2,\infty }(\mathbb {Z}^n)\)  to \({L}_{2,\infty }\) , and, for all \(g\in \ell _{2,\infty }(\mathbb {Z}^n)\) , we have
\(\big \Vert  \lambda (x)g(-i\nabla )\big \Vert _{{L}_{2,\infty }} = \big \Vert  \Phi _x(g)\big \Vert _{{L}_{2,\infty }} & \le \gamma _1(p,q)\gamma _2(p,q)(c_{-}(q)\Vert x\Vert _{L_p})^{1-\theta } (c_+(p)\Vert x\Vert _{L_p})^\theta \Vert g\Vert _{\ell _{2,\infty }}\\ & \le \gamma _1(p,q)\gamma _2(p,q) c_{+}(p)^\theta c_{-}(q)^{1-\theta } \Vert x\Vert _{L_p} \Vert g\Vert _{\ell _{2,\infty }}.\) 

This gives the inequality (REF ).
Here \(\theta \)  is such that \(1/2=(1-\theta )q^{-1}+\theta p^{-1}\) . That is,
\(\theta = \left(\frac{1}{2}-\frac{1}{q}\right) \left(\frac{1}{p}-\frac{1}{q}\right)^{-1}= \frac{p(2-q)}{2(p-q)}, \qquad 1-\theta = \frac{q(p-2)}{2(p-q)}.\) 
Thus, if we denote by \(c(p)\)  the best constant in (REF ), then
\(c(p)\le \gamma _1(p,q)\gamma _2(p,q) c_{+}(p)^{\frac{p(2-q)}{2(p-q)}} c_{-}(q)^{\frac{q(p-2)}{2(p-q)}} \qquad \forall q\in (1,2).\) 
Taking the infimum over \(q\in (1,2)\)  shows that (REF ) is an upper bound for \(c(p)\) . The proof is complete.
